Results 1  10
of
16
The Divisor of Selberg's Zeta Function for Kleinian Groups
 DUKE MATH. J
, 2000
"... We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X ..."
Abstract

Cited by 67 (8 self)
 Add to MetaCart
We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compactified to a manifold with boundary.Ifn is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identified using work of Bunke and Olbrich.
Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds
 Gui05c] [Gui06] [GZ95] [GZ97] [GZ99] [His94] [His00] [Jan79] Colin Guillarmou. Resonances and
, 2005
"... Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N ar ..."
Abstract

Cited by 42 (13 self)
 Add to MetaCart
Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N are at most some poles of finite multiplicity, and that the same 2 property holds for the points of n+1 − N if and only if the metric is ‘even’. On the other 2 hand, there exist some metrics for which R(λ) has an essential singularity on n+1 − N and 2 these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R(λ) approaching an essential singularity.
Dimension Of The Limit Set And The Density Of Resonances For Convex CoCompact Hyperbolic Surfaces.
 Invent. Math
"... this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex cocompact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous ..."
Abstract

Cited by 29 (7 self)
 Add to MetaCart
this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex cocompact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of the limit set of the corresponding Kleinian group. Figure 1. Tesselation by the Schottky group generated by inversions in three symmetrically placed circles each cutting the unit circle in an 110
Inverse Scattering On Asymptotically Hyperbolic Manifolds
 ACTA MATH
, 1998
"... Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown
The wave trace for Riemann surfaces
"... We present a wave group version of the Selberg trace formula for an arbitrary surface of finite geometry. As an application we give a new lower bound on the number of resonances for hyperbolic surfaces. Motivated by recent results we formulate a conjecture on a lower bound for the counting function ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
We present a wave group version of the Selberg trace formula for an arbitrary surface of finite geometry. As an application we give a new lower bound on the number of resonances for hyperbolic surfaces. Motivated by recent results we formulate a conjecture on a lower bound for the counting function of resonances in a strip. 1. Introduction The purpose of this note is to present the Selberg trace formula in terms of the wave group for general surfaces of finite geometry type. The novelty is in allowing infinite volume surfaces and in considering the consequences of the formula for resonances in that case. The study of the Selberg trace formula in this generality was previously conducted by Patterson [10] and the first author [2]. In Sect.2 we present the geometric part of the formula and in Sect.3 we recall some facts about the spectrum and resonances which follow from the more general results of our previous work [4]. Sect.4 is then devoted to a new application to resonance counting. ...
Resonances and scattering poles on asymptotically hyperbolic manifolds
 Math. Res. Lett
"... Abstract. On an asymptotically hyperbolic manifold (X, g), we show that the poles (called resonances) of the meromorphic extension of the resolvent (∆g − λ(n − λ)) −1 coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the point ..."
Abstract

Cited by 15 (8 self)
 Add to MetaCart
Abstract. On an asymptotically hyperbolic manifold (X, g), we show that the poles (called resonances) of the meromorphic extension of the resolvent (∆g − λ(n − λ)) −1 coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the points of n 2 − N. At each λk: = n − k with k ∈ N, the resonance multiplicity 2 m(λk) and the scattering pole multiplicity ν(λk) do not always coincide: ν(λk) − m(λk) is the dimension of the kernel of a differential operator on the boundary ∂ ¯ X introduced by Graham and Zworski; in the asymptotically Einstein case, this operator is the kth conformal Laplacian. 1.
Determinants Of Laplacians And Isopolar Metrics On Surfaces Of Infinite Area
 DUKE MATH. J
, 2001
"... We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zetaregularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.
Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces
 Comment. Math Helv
"... Abstract. For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsionfree, discrete subgroup of SL(2, R ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
Abstract. For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsionfree, discrete subgroup of SL(2, R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [19] and Müller [22] to groups which are not necessarily cofinite. 1.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
"... ..."
A Poisson Summation Formula And Lower Bounds For Resonances In Hyperbolic Manifolds
"... .For convex cocompact hyperbolic manifolds of even dimension n +1,we deriveaPoissontype formula for scattering resonances whichmay be regarded as a version of Selberg's trace formula for these manifolds. Using techniques of Guillop'e and Zworski we easily obtain an O ; R n+1 \Delta lower bound f ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
.For convex cocompact hyperbolic manifolds of even dimension n +1,we deriveaPoissontype formula for scattering resonances whichmay be regarded as a version of Selberg's trace formula for these manifolds. Using techniques of Guillop'e and Zworski we easily obtain an O ; R n+1 \Delta lower bound for the counting function for scattering resonances together with other lower bounds for the counting function of resonances in strips. 1. Introduction In [12], Guillop'e and Zworski proveaPoisson summation formula for the scattering resonances of a hyperbolic surface X with finite geometry, i.e., X consisting of a compact manifoldwithboundary together with a finite number of cusps and funnels. This Poisson summation formula implies a polynomial lower bound on the counting function for scattering resonances and lower bounds for the counting function of scattering resonances in strips. Our purpose here is to generalize both the Poisson summation formula and the lower bounds on resonances to ...